### Rafail Ostrovsky, Bellcore

In 1988, Leighton, Maggs and Rao proved a much celebrated result: that for any network, given any collection of packets with a specified route for each packet, there exists an optimal'' schedule for all these packets. That is, there exists a schedule of the motion of the packets such that at each step, every edge is crossed by at most one packet, and all the packets are delivered to their destinations in $O(C \ + \ D)$ steps, where $C$ is the congestion'' (i.e., the maximum number of paths that share the same edge), and $D$ is the dilation'' (i.e., the length of the longest path). The proof was non-constructive and relied on Lov\'asz Local Lemma. In a followup paper, Leighton, Maggs, and Richa gave a centralized algorithm for finding the schedule. The original paper left open the question whether there exists a constructive distributed on-line'' algorithm with the same optimal performance. Last year, Rabani and Tardos presented a randomized local-control algorithm which with high probability delivers all packets in time $O\left(C \ + \ D \cdot \\ \left((\log^*N)^{O(\log^*N)}\right) + \right.$ $+\left.(\log N)^6\right)$.

In this paper, we show a nearly optimal local control algorithm for this long-standing open problem. That is, we show a randomized local control algorithm which for any network topology delivers all the packets to their destinations in time $O(C \ + \ D \ + \log^{1+\epsilon}N)$ with high probability, where $N$ is the size of the problem, and $\epsilon>0$ is arbitrary. Our result has implications to ATM (Asynchronous Transfer Mode) packet switching algorithms and other applications.

(Joint work with Yuval Rabani)